The impulse acting on a particle analogy gives an incorrect value for the index of refraction so this theory has to be rejected. The direction unit vectors were not the source of the error. If we assume that ê₂ can be decompose in terms of ê₁ and n̂ it is not too difficult to solve for their coefficients.
We should have used the photon's momentum which is proportional to the wave vector, the gradient of the phase. ħ=h/(2π) is the reduced Plank's constant.
Thus we deduce the speed of light will be lower in denser mediums. The ratio of the angles of refraction and reflection equal a constant since the speed of light in the various media is due to the homogeneous nature of the media and so the ratio of the speeds is therefore independent of the angle of incidence. This turned out to be true for the particle impulse theory but we get the inverse of the index of refraction.
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